page_title: Bulk modulus
section_title: Interatomic potential and linear elasticity
hierarchical_section_title: Bulk modulus / Microscopic origin / Interatomic potential and linear elasticity
caption_attribution_description: English: interatomic potential and force
context_page_description: The bulk modulus of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Other moduli describe the material's response to other kinds of stress: the shear modulus describes the response to shear, and Young's modulus describes the response to linear stress. For a fluid, only the bulk modulus is meaningful. For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.
context_section_description: Since linear elasticity is a direct result of interatomic interaction, it is related to the extension/compression of bonds. It can then be derived from the interatomic potential for crystalline materials. First, let us examine the potential energy of two interacting atoms. Starting from very far points, they will feel an attraction towards each other. As they approach each other, their potential energy will decrease. On the other hand, when two atoms are very close to each other, their total energy will be very high due to repulsive interaction. Together, these potentials guarantee an interatomic distance that achieves a minimal energy state. This occurs at some distance a₀, where the total force is zero:
  Where U is interatomic potential and r is the interatomic distance. This means the atoms are in equilibrium.
  To extend the two atoms approach into solid, consider a simple model, say, a 1-D array of one element with interatomic distance of a, and the equilibrium distance is a₀. Its potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal at a₀, The Taylor expansion for this is:
  At equilibrium, the first derivative is 0, so the dominate term is the quadratic one. When displacement is small, the higher order terms should be omitted. The expression becomes:
  Which is clearly linear elasticity.
  Note that the derivation is done considering two neighboring atoms, so the Hook's coefficient is:
  This form can be easily extended to 3-D case, with volume per atom(Ω) in place of interatomic distance.
caption: Interatomic potential and force
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